... linear for large speeds. Our goal is to prove global well-posedness and quantify the long-time decay of the natural energy. Design/methodology/approach We combine the Faedo–Galerkin scheme, Aubin–Lions compactness, and Minty's monotonicity to construct weak solutions and derive an energy identity...

... for our problem. Findings We prove that there exist at least two nontrivial positive weak solutions. Originality/value In particular, it investigates the problem in the presence of singular weights and variable exponents, a case that has not been extensively studied in the existing literature...